Rotated ellipse equation sought

I'm looking for a Cartesian equation for a rotated ellipse.

h is x-koordinate of the center of the ellipse.
k is y-koordinate of the center of the ellipse.
a is the ellipse axis which is parallell to the x-axis when rotation is zero.
b is the ellipse axis which is parallell to the y-axis when rotation is zero.
phi is the rotation angle.

Here is a cartesian equation for a non-rotated ellipse:(How do I "put rotation phi" into this?)

I'd like to have the x and the y in the same equation, and without the parameter t. I think that this means that I want a Cartesian equation for a rotated ellipse, but I'm sorry if I have misunderstood Cartesian here...

Actually, you demonstrated how to derive the parametric equations of a rotated ellipse, which I posted, from the parametric equations of a non-rotated ellipse. I can follow the matrix multiplication, yes, but is it possible to put that together in the way I'm looking for:

Joining x and y in one single equation, and getting rid of that t parameter? If I try to substitute and solve from the parametric equation system, I end up with a mess. There must be a much simpler way.